Given $ m \angle CBD = 2x + 8$, $ m \angle ABC = 2x + 67$, and $ m \angle ABD = 131$, find $m\angle CBD$. $B$ $A$ $D$ $C$
Solution: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Substitute in the expressions that were given for each measure: $ {2x + 67} + {2x + 8} = {131}$ Combine like terms: $ 4x + 75 = 131$ Subtract $75$ from both sides: $ 4x = 56$ Divide both sides by $4$ to find $x$ $ x = 14$ Substitute $14$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 2({14}) + 8$ Simplify: $ {m\angle CBD = 28 + 8}$ So ${m\angle CBD = 36}$.